In this paper we present an algebraic construction of monotonic predicate transformers, using a categorical construction which is similar to the algebraic construction of the integers from the natural numbers. When applied to the category of sets and total functions once, it yields a category isomorphic to the category of sets and relations; a second application yields a category isomorphic to the category of monotonic predicate transformers. This hierarchy cannot be extended further: the category of total functions is not itself an instance of the categorical construction, and can be extended by it only twice.
